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Mirrors > Home > MPE Home > Th. List > dmv | Structured version Visualization version GIF version |
Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
Ref | Expression |
---|---|
dmv | ⊢ dom V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3883 | . 2 ⊢ dom V ⊆ V | |
2 | dmi 5639 | . . 3 ⊢ dom I = V | |
3 | ssv 3883 | . . . 4 ⊢ I ⊆ V | |
4 | dmss 5622 | . . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom I ⊆ dom V |
6 | 2, 5 | eqsstr3i 3894 | . 2 ⊢ V ⊆ dom V |
7 | 1, 6 | eqssi 3876 | 1 ⊢ dom V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 Vcvv 3415 ⊆ wss 3831 I cid 5312 dom cdm 5408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 df-opab 4993 df-id 5313 df-xp 5414 df-rel 5415 df-dm 5418 |
This theorem is referenced by: (None) |
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